So, the burn is an area that is aetherically dead due to being drained hard of it's aether. Subsequently when we head to the Allagan ruins there, we find Allagan weapons that have the Charred suffix, meaning likely that they have been affected by being left in the Burn.
How are they still running then? The most likely side effect of remaining there would be any stored aether draining out of you like being in a vacuum, and as they run on aether the automatons would likely be either dead or severely drained, which doesn't seemt to be the case.
Also, the Burn and Charred seem like weird adjectives to use. Those seem more at home with the effects of too much aether in an area, not too little.